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Full text of "Men Of Mathematics"

MEN OF MATHEMATICS would give him all the invariant expressions of the kind described* To avoid lengthy explanations the problem has been stated in terms of equations; actually it was attacked otherwise, but this is of no importance here. As this question of invariance is fundamental in modern scientific thought we shall give three further illustrations of what it means, none of which involves any symbols or algebra. Imagine any figure consisting of intersecting straight lines and curves drawn on a sheet of paper. Crumple the paper in any way you please without tearing it, and try to think what is the most obvious property of the figure that is the same before and after crumpling. Do the same for any figure drawn on a sheet of rubber, stretching but not tearing the rubber in any compli- cated manner dictated by whim. In this case it is obvious that sizes of areas and angles, and lengths of lines, have not remained 'invariant*. By suitably stretching the rubber the straight lines may be distorted into curves of almost any tortuosity you like, and at the same time the original curves - or at least some of them - may be transformed into straight lines. Yet sometHing about the whole figure has remained unchanged; its very sim- plicity and obviousness might well cause it to be overlooked, Ttiis is the order of the points on any one of the lines of the figure which mark the places where other lines intersect the given one. Thus, if moving the pencil along a given line from A to C, we had to pass over the point B on the line before the figure was distorted, we shall have to pass over B in going from A to C after distortion, The order (as described) is an invariant under the particular transformations which crumpled the sheet of paper into a crinkly ball, say, or which stretched the sheet of rubber.* This illustration may seem trivial, but anyone who has read a non-mathematical description of the intersections of 'world- lines* irbgeneral relativity, and who recalls that an intersection of two such lines marks a physical 'point-eoertf, will see that what we have been discussing is of the same stuff as one of our pictures of the physical universe. The mathematical machinery powerful enough to handle such complicated 'transformations* and actually to produce the invariants was the creation of 430